Monday, June 08, 2015

Mismatched Estimation in Large Linear Systems

Mismatched estimation in compressed sensing:
Many scientific and engineering problems can be approximated
as linear systems, where the input signal is modeled as a
vector, and each observation is a linear combination of the
entries in the input vector corrupted by white Gaussian noise.
In joint work with
Yanting Ma and
Ahmad Beirami,
the input signal is modeled as a realization of
a vector of independent and identically distributed random
variables. The goal is to estimate the input signal such that
the mean square error (MSE), which is the Euclidean distance
between the estimated signal and the true input signal averaged
over all possible realizations of the input and the observation,
is minimized. It is well-known that the best possible MSE,
minimum mean square error (MMSE), can be achieved by computing
conditional expectation, which is the mean or average value of
the input given the observation vector, where the true
distribution of the input is used.
However, the true distribution is usually not known exactly in
practice, and so conditional expectation is computed with a
postulated distribution that differs from the true distribution;
we call this procedure mismatched estimation, and it
yields an MSE that is higher than the MMSE. We are
interested in characterizing the excess MSE (EMSE) above the
MMSE due to mismatched estimation in large linear systems,
where the length of the input and the number of observations
grow to infinity together, and their ratio is fixed......"

We study the excess mean square error (EMSE) above the minimum mean square
error (MMSE) in large linear systems where the posterior mean estimator (PME)
is evaluated with a postulated prior that differs from the true prior of the
input signal. We focus on large linear systems where the measurements are
acquired via an independent and identically distributed random matrix, and are
corrupted by additive white Gaussian noise (AWGN). The relationship between the
EMSE in large linear systems and EMSE in scalar channels is derived, and closed
form approximations are provided. Our analysis is based on the decoupling
principle, which links scalar channels to large linear system analyses.
Numerical examples demonstrate that our closed form approximations are
accurate.